Geometric Algebra and Möbius Sphere Geometry as a Basis for Euclidean Invariant Theory

Physicists have traditionally described their systems by means of explicit parametrizations of all their possible individual configurations. This makes a local description of the motion of the system relatively simple, but provides little insight into the global properties of its solution space. Geometers, on the other hand, tend to describe their systems implicitly in terms of their invariant geometric properties. This approach has the substantial advantage of enabling them to deal with entire sets of configurations simultaneously, and renders every theorem global in scope. In physical problems, an analogous approach would use invariants of the Lie group underlying the dynamical system in question, e.g. angular momentum in the case of rotationally invariant dynamics.